A Note on the Pairing Computation Using Normalized Miller Functions

Abstract

This paper considers the normalization of Miller functions for computing “point-evaluation” pairings on an elliptic curve E over a finite field $\\GF_{q}$, where the characteristic of $\\GF_{q}$ is neither 2 nor 3. It is shown that the normalized Miller functions for computing point-evaluation pairings on $\\G_2 \\ imes \\G_1$ when (i) the embedding degree k is even, or (ii) 3|k and E/$\\GF_{q}$(q≡1(mod 3) is a curve of the form Y2=X3+b. Thus, there is no need to consider the normalization for computing pairings on many pairing-friendly elliptic curves.